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Theorem nss 3988
Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
nss 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nss
StepHypRef Expression
1 exanali 1866 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
2 dfss2 3912 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2xchbinxr 335 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ 𝐴𝐵)
43bicomi 223 1 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1540  wex 1786  wcel 2110  wss 3892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909
This theorem is referenced by:  grur1  10577  psslinpr  10788  reclem2pr  10805  mreexexlem2d  17352  prmcyg  19493  filconn  23032  alexsubALTlem4  23199  wilthlem2  26216  shne0i  29806  erdszelem10  33158  fundmpss  33736  ntrneineine1lem  41664  nssrex  42606  nssd  42625  nsssmfmbf  44282
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